I wish I better understood WHY this was necessary.
At one point my BBB "just worked". When it was plugged into my desktop computer via USB (Windows 7 machine) the BBB had internet connectivity just fine. After working to solve other issues it stopped. In any case, it works again with the following steps:
On the Windows 7 computer:
Control Panel -> Network and Internet -> Network Connections
Right-click your internet connection, Properties, Sharing Tab
Click "Allow other network users to connect thru this computer's Internet connection"
Select the BeagleBone Black's network connection under "Home networking connection:"
Here's the trick: when I did this, Windows assigned a static IP to the BBB connection. Why? No idea.
You have to go back to that connection and reset it to DHCP.
Control Panel -> Network and Internet -> Network Connections
Right-click the BeagleBone Black's connection, Properties
Click Internet Protocol Version 4 (TCP/IPv4), Click Properties
To make this "stick" after a reboot of the BBB, you need to add these commands to the ethernet startup script on the BBB (/usr/bin/g-ether-load.sh).
NOTE: You CANNOT simply add these commands to the bottom of that script file. They must go before the udhcpd starts. It does not return control to the script so adding them at the bottom of the file does nothing for you. My script file now looks like this: